Arbitrary Lagrangian-Eulerian (ALE) based finite element formulation uses a computational system that is not fixed in space (e.g. Eulerian-based finite element formulations) or attached to material (e.g. Lagrangian-based finite element formulations). ALE based finite element simulations can alleviate many of the drawbacks that the traditional Lagrangian-based and Eulerian-based finite element simulations have. ALE techniques can be applied to many engineering problems, for example, fluid-structure interaction, coupling of multi-physics fields with multi-materials (moving boundaries and interfaces), metal forming/cutting, casting, and the likes.
When using the ALE technique in engineering simulations, the computational mesh inside the domains can move arbitrarily to optimize the shapes of elements, while the mesh on the boundaries and interfaces of the domains can move along with materials to precisely track the boundaries and interfaces of a multi-material system.
In a time marching simulation using ALE based finite element analysis (FEA), simulated responses are obtained in two solution phases or cycles (Lagrangian and advection) at each time step. First, in Lagrangian phase, responses of the FEA mesh model in form of material flux are computed. Nodes of the FEA mesh are moved accordingly. Next, in the advection phase, the computed material flux are mapped back to original undeformed mesh by letting computed material flux (i.e., deformed portion measured in volume) move out of a donor into one or more receptors.
In order to obtain better simulated responses, the FEA mesh needs to be refined at locations of interest. One of the problems associated with refining ALE elements is related to mapping of the computed results, which has been problematic in prior art approaches. It would therefore be desirable to have improved methods and systems for refining ALE elements in a time-marching simulation using ALE based FEA.